QUADRATIC FREQUENCY DEPENDENCE OF AN AC QHR DEVICE

M. E. Cage

National Institute of Standardsand TechnologyGaithersburg,MD 20899-8172

Abstract

An equivalent electrical circuit model of an acquantized Hall resistance device externallyconnected in quadruple series predicts a

quadratic fre~uency dependence no more than-4 x 10-9/kHz. This agrees with experimentwhen indirectly comparing that device with acalculable quadrifilar resistor in interchanged 1:1ratios using a four-terminal-pair bridge.

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Introduction

AC quantized Hall resistance (ac QHR)standards have linear and quadratic frequencydependences, as do the reference resistors withwhich they are compared, and the acmeasurement bridges themselves [1]. This paperdeals only with the quadratic dependences. Thebridge quadratic dependence is dominant [1] andmust be removed. Then either a referenceresistor with a calculable quadratic dependenceis required to extract its ac-dc difference [2] andthereby determine the quadratic dependence ofthe ac QHR standard, or the quadraticdependence of the ac QHR standard itself mustbe determined [3]. This experiment does all ofthe above.

Quadrifilar (Gibbings) resistors [2] are oftenemployed as calculable reference resistors toextract the ac-dc differences. Uniformtransmission line models can calculate quadraticfrequency effects due to parasitic inductancesand capacitances, and other models calculatequadratic dependences due to skin effects andeddy currents in the resistor wires andsurrounding shield.

Quadrifilar resistance values change because ofinstabilities in the thin wires. So rather thandirectly compare an ac QHR device with aquadrifilar resistor,. we found it easier tocompare a quadrifilar resistor with a stable wire-wound resistor without having to bother with asuperconducting magnet and cyrogens, and thencompare that stable wire-wound resistor with anac QHR device using the magnet.

Reference [1] describes those comparisons indetail, demonstrates that all dc QHR guidelineproperties and all dc and ac QHR values can bemeasured in a single cooldown without changing

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sample probe leads at the device, and shows thatquadruple-series-connected ac QHR valuesmeasured in the four-terminal-pair modeconverge to the dc QHR value. The next sectionsummarizes that experiment.

EXDeriment

The only ac QHR device available was aGaAs/AIGaAs heterostructure we labeled acl. Itwas mounted on a custom-built header using100 J.1m diameter platinum wires to avoidvibrational effects of ac currents in a magneticfield. The header had a ground plane over mostof its back surface to minimize wire-to-wirecapacitances at the device (which is critical toreduce quadratic frequency dependences whenusing quadruple-series connections [3D. Thedevice was homogeneous, but had potential padcontact resistances as large as 7,644 n. That7,644 n contact exhibited Corbino-likebehavior: its resistance increased with decreasingcurrent at small currents due to isolated spikesdiffused into the two-dimensional electron fluid.This behavior created problems: static voltagesinduced when changing connections duringbalances sometimes required minutes or hours todecay, causing the QHR values to fluctuate. Thatlimited the tyP.e A 1cr relative uncertainties toII parts in 107, which was frustrating with 5parts in 109measurement resolutions at 20 J.1A.

A 12,906.4 n wire-wound resistor labeled12.9WWl was assembled from resistorcomponents made of wire wound on mica cardsand placed in a shielded container. It wasdesigned for minimum capacitances-to-shieldand placed in an oil bath.

A quadrifilar resistor, also of nominal12,906.4 n value and labeled 12.9QF1, had aself-contained air bath and was thermally-laggedwith additional insulation. Its resistance couldunpredictably change a few parts in 107 overseveral hours, necessitating computationaladjustment to a reference frequency that wechose as 1,592 Hz.

The three dc single-series 12,906.4 n i = 2quantized Hall resistances of QHR device acl,and its externally-connected dc quadruple-seriesresistance, were compared with the 12,906.4 nwire-wound reference resistor 12.9WWl at29 J.1A and 1.6 K using an automated

U.S. Government work not protected by U.S. copyright.

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measurement system that reverses currents,exchanges positions of two digital multimeters,and exchanges the positions of ac 1 and12.9WW1. The four ac1/12.9WWl dc ratiovalues were consistent to within ::!:1parts in 107;the large fluctuations were the result of theCorbino effect.

The quadruple-series-connected ac quantizedHall resistance was then compared with wire-wound resistor 12.9WWl at 20 ~A and 1.5 K ininterchanged 1:1 ratios from 700 Hz to 5,500 Hzusing a four-terminal-pair (4TP) multifrequencytransformer bridge and five in-phase and fiveout-of-phase balances. Quadrifilar resistor12.9QFl was also compared with wire-woundresistor 12.9WWl over the same frequencyrange.

Corbino-like behavior of the 7,644 Q potentialcontact again limited the ac QHR measurementaccuracies to ::!:1 parts in 107. Interchanged 1:1ratio measurements allowed removal of thedominant (-12.07::!: 0.10) x 1O-7fkHz2 bridgequadratic dependence from the acll12.9WW1and 12.9QFlI12.9WWl ratios. The quadraticfrequency dependence of the interchangedacll12.9WW1 ratio was then (-1.15:!: 0.10) x10-7fkHz2, and the 12.9QFlI12.9WW1 ratiodependence was (-1.02:!: 0.10) x 1O-7fkHz2.Therefore the indirectly measured acl/12.9QFlquadratic dependence is (-1.3::!: 1.4) x 1O-81kHz2.

Equivalent Circuit Model

Jeffery, Elmquist, and Cage [4] showed that theRicketts and Kemeny equivalent electrical circuitmodel [5] represents multi-series-connected dcQHR devices to 1 part in 109'accuracies. ThatRicketts/Kemeny model [5] lies at the heart ofthe Cage/Jeffery/Matthews equivalent electricalcircuit representation [3] of an ac QHR standardwith two external quadruple-series connectionswhen measured under 4TP balance conditions.

Most of the 50 parasitic impedances in theCage/Jeffery/Matthews model [3] were found at4.2 K by placing shorted and open-circuitedback-grounded headers in the sample probebefore mounting and cooling ac QHR deviceac 1. DC longitudinal resistances in the modelwere measured at the Vx minima of the i = 2QHR plateau region to 1 part in 109 accuracyusing an automated potentiometeric system.

Results

The predicted quadratic frequency dependenceof ac QHR device acl is at most -4 x 10-9fkHz2(and probably much smaller) in this equivalentcircuit model when using the 50 measuredparasitic impedancesand an exact solution of the45 coupled equations in the model [3]. This

prediction implies that the (-1.3 ::!:1.4)x 10-8fkHZ2 quadratic dependence of theinterchanged ac1/12.9QFl ratio is primarily dueto the quadrifilar resistor 12.9QFl, and that12.9QF1's quadratic dependence is(+ 1.3 ::!:1.4) x 10-8fkHZ2. That value isconsistent with the + 1.34 x 1O-8/kHz2 valueobtained from the transmission line, eddycurrent, and skin effect calculations of B. M.Wood [6] for our quadrifilar resistor.

Conclusions

It appears that quadruple-series-connected acQHR devices mounted on headers that minimizethe wire-to-wire capacitances have very smallquadratic frequency dependences whenmeasured in interchanged 1:1 ratios under 4TPbalance conditions. Repeating this experimentwith a good device would improve themeasurement accuracies about an order ofmagnitude, and would rigorously test thequadruple-series-connected equivalent circuitmodel predictions [3] for ac QHR devices andthe ac-dc difference calculations [2,6] forquadrifilar resistors.

Acknowledl!ments

B. M. Wood at the NRC, Canada supplied theprogram that calculates the quadratic frequencydependences of quadrifilar resistors.

References

[1] M. E. Cage, S. H. Shields, and A. Jeffery,"Initial NIST ac QHR Measurements,"to bepublished.

[2] D. L. H. Gibbings, "A Design for Resistorsof Calculable AC/DC Resistance Ratio,"Proc. lEE, Vol. 110,pp. 335-347,1963.

[3] M. E. Cage, A. Jeffrey, and 1. Matthews,"Equivalent Electrical CircuitRepresentations of AC Quantized HallResistance Standards," J. Res. Natl. Inst.Stand. Technol., Vol. 104 (6), pp. 529-556,1999.

[4] A. Jeffery, R. E. Elmquist, and M. E. Cage,"Precision Tests of a Quantum Hall EffectDevice DC Equivalent Circuit usingDouble-Series and Triple-SeriesConnections," J. Res. Natl. Inst. Stand.Technol.,Vol. 100(6), pp. 677-685, 1995.

[5] B. W. Ricketts and P. C. Kemeny,"Quantum Hall Effect Devices as CircuitElements," J. Phys. D: Applied Phys., Vol.21, pp. 483-487, 1988.

[6] B. M. Wood, NRC, Canada, privatecommunication.

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